U.S. patent number 5,298,972 [Application Number 07/755,931] was granted by the patent office on 1994-03-29 for method and apparatus for measuring polarization sensitivity of optical devices.
This patent grant is currently assigned to Hewlett-Packard Company. Invention is credited to Brian L. Heffner.
United States Patent |
5,298,972 |
Heffner |
March 29, 1994 |
Method and apparatus for measuring polarization sensitivity of
optical devices
Abstract
An instrument includes a polarized optical source for producing
three sequential predetermined states of polarization of a light
beam and an optical polarization meter for measuring the
polarization of a portion of the light beam transmitted by or
reflected from an optical network by splitting it into four beams,
passing three of the beams through optical elements, measuring the
transmitted intensity of all four beams, and calculating Stokes
parameters. The light beam enters the optical polarization meter
through a single-mode optical fiber that acts as a spatial filter
for controlling the position and alignment of the beam with respect
to the optical elements. The distortion of the light beam
polarization caused by this optical fiber is corrected by
introducing two different linearly polarized light beams and
measuring Stokes parameters which are used to construct a
calibration matrix that is inverted and multiplied times measured
Stokes parameters of subsequent measurements to yield true Stokes
parameters. The three sequential predetermined states of
polarization yield three corresponding Jones input vectors, and the
Stokes parameters for the responses of the optical network are
converted to three Jones output vectors. A Jones matrix for the
optical network to within a complex constant is then computed from
the Jones input and output vectors. Relative polarization
sensitivity can be determined from this matrix for the optical
network. The relative distortion caused by the optical network can
be corrected by multiplying by the inverse of the matrix during
later measurements through the optical network. Additionally, power
measurements on the optical network and a substituted optical
through enable absolute determinations and corrections.
Inventors: |
Heffner; Brian L. (Redwood
City, CA) |
Assignee: |
Hewlett-Packard Company (Palo
Alto, CA)
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Family
ID: |
25041290 |
Appl.
No.: |
07/755,931 |
Filed: |
September 6, 1991 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
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601613 |
Oct 17, 1990 |
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468397 |
Jan 22, 1990 |
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Current U.S.
Class: |
356/364 |
Current CPC
Class: |
G01M
11/00 (20130101); G01J 4/04 (20130101) |
Current International
Class: |
G01J
4/00 (20060101); G01M 11/00 (20060101); G01J
4/04 (20060101); G01J 004/04 () |
Field of
Search: |
;356/364,365,366,367
;250/225 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
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0249235 |
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Jun 1967 |
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EP |
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871113932 |
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Aug 1987 |
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EP |
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893074776 |
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Jul 1988 |
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EP |
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0352133 |
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Jul 1989 |
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EP |
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2570186 |
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Sep 1984 |
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FR |
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1343253 |
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Oct 1987 |
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SU |
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WO86/07631 |
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Dec 1986 |
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WO |
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Other References
"Polarization Stabilization on Single-Mode Fiber", R. Ulrich, Appl.
Phys. Lett. 35(11), Dec. 1, 1979. .
"Polarization Fluctuations of Optical Fibre Submarine Cable in
6000m Deep Sea Trial", Electronics Letters, May 12, 1988. .
"Polarization Fluctuations in a 147 km Undersea Lightwave Cable
During Installation", Electronics Letters, Oct. 8, 1987. .
"General Analysis and Optimization of the Four-Detector
Photopolarimeter", R. A. M. Azzam, I. M. Elminyawi, and A. M.
El-Saba, J. Opt. Soc. Am. A., May 5, 1988. .
"Real-Time Heterodyne Fiber Polarimetry with Narrow-and-Broad-Band
Sources", Riccardo Calvani, Renato Caponi, and Francesco
Cisternino, Journal of Lightwave Technology, Jul. 1986. .
"Construction, Calibration and Testing of a Four-Detector
Photopolarimeter", R. M. A. Azzam, E. Masetti, I. M. Elminyawi, and
F. G. Grosz, Rev. Sci. Instrum. Jan. 1988. .
"Determination of the Ellipsometric Characteristics of Optical
Surfaces Using Nanosecond Laser Pulses", Edward Collett, Surface
Science96 (1980) 156-167. .
"Ellipsometry with Pulsed Tunable Laser Sources", Dill et al., IBM
Technical Disclosure Bulletin, vol. 19, No. 4 (Sep. 1976), pp.
1487-1489. .
S. R. Cloude, "Group Theory and Polarization Algebra", OPTIK, Dec.
1986, No. 1, Stuttgart, W. Germany, p. 26-36..
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Primary Examiner: Rosenberger; Richard A.
Parent Case Text
CROSS-REFERENCE TO RELATED APPLICATION
This is a continuation-in-part of copending U.S. Pat. application
Ser. No. 07/601,613 filed Oct. 17, 1990 now abandoned which in turn
is a continuation-in-part of 07/468,397 filed Jan. 22, 1990 now
abandoned.
Claims
What is claimed is:
1. A method for calibrating an optical polarization meter to
compensate for any polarization distortion caused by optical
elements, which polarization distortion leads to transmission loss
that is independent of polarization, the method comprising the
steps of:
transmitting a first incident polarized light beam from a polarized
optical source through the optical elements to the meter;
measuring a set of Stokes parameters of the first incident
beam;
computing a first normalized Stokes vector according to the Stokes
parameters of the first incident beam;
transmitting a second incident polarized light beam from the
polarized optical source through the optical elements to the meter,
the second incident beam having a polarization different from that
of the first incident beam;
measuring a set of Stokes parameters of the second incident
beam;
computing a second normalized Stokes vector according to the Stokes
parameters of the second incident beam;
computing a set of calibration factors according to the normalized
Stokes vectors; and
correcting polarization parameters of an unknown light beam as
measured by the meter according to the calibration factors to
provide a calibrated polarization measurement.
2. A method according to claim 1 wherein the optical elements
comprise an optical spatial filter of the optical polarization
meter.
3. A method according to claim 1 wherein computing the calibration
factors comprises:
computing a first calibration vector by scaling each element of the
first normalized Stokes vector by the magnitude of the first
normalized Stokes vector;
computing a second calibration vector by scaling each element of
the second normalized Stokes vector by the magnitude of the second
normalized Stokes vector; and
computing a third calibration vector by taking the vector cross
product of the second calibration vector and the first calibration
vector and then scaling each element of the third calibration
vector by the magnitude of the third calibration vector.
4. A method according to claim 3 wherein computing the calibration
factors comprises:
forming a first matrix having three columns each comprising
elements that are equal to elements of one of the calibration
vectors;
forming a second matrix which includes the first matrix; and
taking the inverse of the second matrix to obtain the calibration
factors.
5. A method according to claim 1 wherein the first and second
incident polarized light beams are linearly polarized and the
relative angle between the direction of polarization of the first
incident polarized light beam and the second incident polarized
light beam is 45 degrees.
6. An instrument for measuring polarization sensitivity of an
optical network, the instrument comprising:
polarized optical source means that sequentially generates three
predetermined states of polarization of a light beam and impinges
the light beam having each of the three predetermined polarization
states onto the optical network;
optical polarization measurement means that receives a portion of
each of the three predetermined polarization states of the light
beam that is one of a) transmitted by and b) reflected from the
optical network and measures the polarization states produced by
the effect of the optical network on each of the three
predetermined polarization states of the beam; and
means for computing a ratio of maximum and minimum values from the
measured polarization states produced by the effect of the optical
network on each of the three predetermined polarization states of
the beam to thereby provide a measurement of the polarization
sensitivity of the optical network.
7. The instrument as in claim 6 wherein the polarized optical
source means comprises an optical source for generating a beam
light, the optical source having an output, and a polarization
synthesizer having an input connected to the output of the optical
source, the polarization synthesizer comprising optical elements
for producing the three predetermined states of polarization of the
light beam in response to the beam of light received from the
optical source.
8. The instrument as in claim 7 wherein the optical elements of the
polarization synthesizer comprise a zero-degree linear polarizer, a
60-degree linear polarizer, and a 120-degree linear polarizer
sequentially inserted into a path of the beam of light received
from the optical source.
9. An instrument for measuring polarization sensitivity of an
optical network, the instrument comprising:
polarized optical source means that sequentially generates three
predetermined states of polarization of a light beam and impinges
the light beam having each of the three predetermined polarization
states onto the optical network;
optical polarization measurement means that receives a portion of
each of the three predetermined polarization states of the light
beam that is one of a) transmitted by and b) reflected from the
optical network and measures the polarization states produced by
the effect of the optical network on each of the three
predetermined polarization states of the beam, the optical
polarization measurement means comprising:
an optical spatial filter which receives and filters each of the
three predetermined polarization states of the portion of the light
beam from the optical network;
means for splitting the filtered portion of the light beam into
four partial beams;
a first optical element located in the path of a first one of the
partial beams and imparting a first polarization thereto;
a second optical element located in the path of a second one of the
partial beams and imparting a second polarization thereto;
a third optical element located in the path of a third one of the
partial beams and imparting a third polarization thereto;
four photodetectors, each photodetector receiving a different one
of the four partial beams and providing a signal indicative of the
intensity of that partial beam; and
means for computing the polarization of the incident beam of light
from the signals provided by the photodetectors; and
means for computing a ratio of maximum and minimum values from the
measured polarization states produced by the effect of the optical
network on each of the three predetermined polarization states of
the beam to thereby provide a measurement of the polarization
sensitivity of the optical network.
10. The instrument as in claim 9 wherein the polarized optical
source means comprises an optical source for generating a beam of
light, the optical source having an output, and a polarization
synthesizer having an input connected to the output of the optical
source, the polarization synthesizer comprising optical elements
for producing the three predetermined states of polarization of the
light beam in response to the beam of light received from the
optical source.
11. The instrument as in claim 10 wherein the optical elements of
the polarization synthesizer comprise a zero-degree linear
polarizer, a 60-degree linear polarizer, and a 120-degree linear
polarizer sequentially inserted into a path of the beam of light
received from the optical source.
12. A method for measuring the polarization sensitivity of an
optical device under test based on one of a) transmission and b)
reflection responses of said optical device under test, the method
comprising:
producing a polarized light beam having three sequential
predetermined states of polarization, corresponding to three input
optical electric field Jones vectors;
impinging the light beam on the optical device under test;
measuring one of the a) transmission and b) reflection responses of
the optical device under test to the light beam for each of the
three sequential predetermined polarization states by measuring the
Stokes parameters of the responses of the optical device under
test;
computing Stokes vectors from the Stokes parameters;
converting the Stokes vectors to output optical electric field
Jones vectors;
computing a Jones matrix to within a complex constant for the
optical device under test from the Jones input and output vectors;
and
computing a ratio of the maximum and minimum values of the one of
the a) transmission and b) reflection responses to provide a
measurement of the polarization sensitivity of the optical device
under test in relative terms from the matrix.
13. A method according to claim 12, further comprising the step of
computing states of polarization corresponding to the at least one
of the maximum and minimum values of the one of the a) transmission
and b) reflection responses of the optical device under test.
14. A method for calibrating an instrument to correct for
distortion of polarization state caused by an optical network which
is not completely polarizing, comprising the steps of:
producing a polarized light beam having three sequential
predetermined states of polarization, corresponding to three input
optical electric field Jones vectors;
impinging the light beam on the optical network;
measuring one of the a) transmission and b) reflection responses of
the optical network to the light beam for each of the three
sequential predetermined polarization states by measuring the
Stokes parameters of the responses of the optical network;
computing Stokes vectors from the Stokes parameters for the optical
network;
converting the Stokes vectors correlated to the responses of the
optical network to output optical electric field Jones vectors;
computing a Jones matrix to within a complex constant for the
optical network from the Jones input and output vectors;
computing the inverse of the matrix for the optical network;
and
correcting parameters of an unknown light beam as measured by the
instrument according to the inverse of the matrix for the optical
network to provide a calibrated measurement of said parameters
whereby said parameters are correctly measured notwithstanding any
distortion of the state of polarization caused by the optical
network.
15. A method according to claim 14, further comprising the steps
of:
inserting an optical device under test into the path of the light
beam;
measuring one of the a) transmission and b) reflection responses of
the optical network and the optical device under test to the light
beam for each of the three sequential predetermined polarization
states by measuring the Stokes parameters of the responses of the
optical network and the optical device under test;
computing second Stokes vectors from the Stokes parameters for the
optical network and the optical device under test;
converting the second Stokes vectors correlated to the responses of
the optical network and the optical device under test to second
output optical electric field Jones vectors;
computing a Jones matrix to within a complex constant for the
optical device under test from the Jones input vectors and second
Jones output vectors after multiplication of the second Jones
output vectors by the inverse of the matrix for the optical
network; and
computing a ratio of maximum and minimum values of the one of a)
transmission and b) reflection responses of the optical device
under test to provide a measurement of the polarization sensitivity
of the optical device under test in relative terms from the matrix
for the optical device under test.
16. A method according to claim 15, further comprising the step of
computing states of polarization corresponding to the at least one
of the maximum and minimum values of the one of the a) transmission
and b) reflection responses of the optical device under test.
17. A method for measuring the polarization sensitivity of an
optical device under test based on one of a) transmission and b)
reflection responses of the optical device under test, the method
comprising the steps of:
producing a polarized light beam having three sequential
predetermined states of polarization, corresponding to three input
optical electric field Jones vectors;
impinging the light beam on an optical through;
measuring one of the a) transmitted and b) reflected power of the
optical through to the light beam for at least one of the three
sequential predetermined polarization states;
substituting the optical device under test for the optical
through;
impinging the light beam on the optical device under test;
measuring one of the a) transmission and b) reflection responses of
the optical device under test to the light beam for each of the
three sequential predetermined polarization states by measuring the
Stokes parameters of the responses of the optical device under
test;
measuring one of the a) transmitted and b) reflected power of the
optical device under test to the light beam for at least one of the
three sequential predetermined polarization states;
computing Stokes vectors from the Stokes parameters for the optical
device under test;
converting the Stokes vectors correlated to the responses of the
optical device under test to output optical electric field Jones
vectors;
computing a Jones matrix for the optical device under test from the
Jones input and output vectors and the power measurements obtained
with the optical through and the optical device under test; and
providing a measurement of the polarization sensitivity of the
optical device under test in absolute terms from the Jones matrix
by computing a ratio of the maximum and minimum values of the one
of the a) transmission and b) reflection responses.
18. A method according to claim 17, further comprising the step of
computing states of polarization corresponding to the at least one
of the maximum and minimum values of the one of the a) transmission
and b) reflection responses of the optical device under test.
19. A method for calibrating an instrument to correct for
distortion of polarization state caused by an optical network which
is not completely polarizing, comprising the steps of:
producing a polarized light beam having three sequential
predetermined states of polarization, corresponding to three input
optical electric field Jones vectors;
impinging the light beam on an optical through;
measuring one of the a) transmission and b) reflection responses of
the optical through to the light beam for each of the three
sequential predetermined polarization states by splitting the light
beam from the optical through to be measured into four beams,
passing three of the beams through optical elements, and measuring
the intensities of all four beams;
measuring one of the a) transmitted and b) reflected power of the
optical through to the light beam for each of the three sequential
predetermined polarization states;
substituting the optical network for the optical through;
impinging the light beam on the optical network;
measuring one of the a) transmission and b) reflection responses of
the optical network to the light beam for each of the three
sequential predetermined polarization states by measuring the
Stokes parameters of the responses of the optical network;
measuring one of the a) transmitted and b) reflected power of the
optical network to the light beam for at least one of the three
sequential predetermined polarization states;
computing Stokes vectors from the Stokes parameters for the optical
network;
converting the Stokes vectors correlated to the responses of the
optical network to output optical electric field Jones vectors;
computing a Jones matrix for the optical network from the Jones
input and output vectors and the power measurements obtained with
the optical through and the optical network;
determining the inverse of the Jones matrix for the optical
network; and
correcting responses represented by additional output optical
electric field Jones vectors obtained during subsequent
polarization state measurements according to the inverse of the
Jones matrix for the optical network to provide calibrated absolute
measurements of said subsequent polarization states notwithstanding
any distortion of the state of polarization caused by the optical
network.
20. A method according to claim 19, further comprising the steps
of:
inserting an optical device under test into the path of the light
beam;
measuring one of the a) transmission and b) reflection responses of
the optical network and the optical device under test to the light
beam for each of the three sequential predetermined polarization
states by measuring the Stokes parameters of the responses of the
optical network and the optical device under test;
computing second Stokes vectors from the Stokes parameters for the
optical network and the optical device under test;
converting the second Stokes vectors correlated to the responses of
the optical network and the optical device under test to second
output optical electric field Jones vectors;
computing a Jones matrix to within a complex constant for the
optical device under test from the Jones input vectors and the
second Jones output vectors after multiplication of the second
Jones output vectors by the inverse of the matrix for the optical
network; and
computing a ratio of maximum and minimum values of the one of a)
transmission and b) reflection responses of the optical device
under test to provide a measurement of the polarization sensitivity
of the optical device under test in relative terms from the matrix
for the optical device under test.
21. A method according to claim 20, further comprising the step of
computing states of polarization corresponding to the at least one
of the maximum and minimum values of the one of the a) transmission
and b) reflection responses of the optical device under test.
22. A method according to claim 19, further comprising the steps
of:
connecting the optical through to the optical network;
measuring one of the a) transmitted and b) reflected power of the
optical through and the optical network to the light beam for at
least one of the three sequential predetermined polarization
states;
substituting an optical device under test for the optical
through;
impinging the light beam on the optical device under test;
measuring one of the a) transmission and b) reflection responses of
the optical network and the optical device under test to the light
beam for each of the three sequential predetermined polarization
states by measuring the Stokes parameters of the optical network
and the optical device under test;
measuring one of the a) transmitted and b) reflected power of the
optical network and the optical device under test to the light beam
for at least one of the three sequential predetermined polarization
states;
computing second Stokes vectors from the Stokes parameters for the
optical network and the optical device under test;
converting the second Stokes vectors correlated to the responses of
the optical network and the optical device under test to second
output optical electric field Jones vectors;
computing a Jones matrix for the optical device under test from the
Jones input vectors and the second Jones output vectors after
multiplication of the second Jones output vectors by the inverse of
the matrix for the optical network and the power measurements
obtained with the optical through, the optical network, and the
optical device under test; and
providing a measurement of the polarization sensitivity of the
optical device under test in absolute terms from the matrix for the
optical device under test by computing a ratio of the maximum and
minimum values of the one of a) transmission and b) reflection
responses.
23. A method according to claim 22, further comprising the step of
computing states of polarization corresponding to the at least one
of the maximum and minimum values of the one of the a) transmission
and b) reflection responses of the optical device under test.
Description
BACKGROUND OF THE INVENTION
This invention relates to the field of electronic instruments for
measuring the polarization state of a beam of light and, more
particularly, to such instruments that are capable of detecting
effects on the polarization state of an incident light beam caused
by an optical device under test (i.e., an optical system,
subsystem, or component). Specifically, one embodiment of the
invention provides a method and apparatus for impinging a light
beam having predetermined states of polarization on an optical
device under test to ascertain a response characteristic of the
optical device to different polarization states and determining the
polarization sensitivity of the optical device. One embodiment of
the invention provides automatic polarization sensitivity
determination to ascertain, for example, maximum and minimum
transmission (or maximum and minimum reflection) of an optical
device under test in response to the different possible states of
polarization of an incident light beam, and the respective
polarization states at which the maximum and the minimum
transmission (or maximum and minimum reflection) occur.
There are various known techniques for measuring the polarization
state of a light beam. The conventional technique for measuring the
polarization state of a light beam is to align a waveplate and a
linear polarizer in the optical path of the beam. The waveplate is
configured to be rotatable about the optical axis, and is typically
a quarter-wave plate. An optical sensor, such as a photodetector,
is positioned downstream to measure the intensity of light
transmitted by the waveplate and the polarizer.
In operation, the waveplate is sequentially rotated to a plurality
of angular positions about the optical axis relative to the linear
polarizer, and the transmitted light intensity is measured at each
angular position by the photodetector. Intensity measurements at a
minimum of four different angular positions are required for a
determination of the state of polarization of the light beam. As is
well-known, the polarization state of the light beam can be
computed from these intensity measurements.
This technique has the drawback that it requires mechanical
movement of the waveplate. Therefore, the speed of measurement of
the polarization state is limited by the speed with which the
waveplate can be rotated, and, in the case that the waveplate is
rotated manually, measurement of the polarization state is
time-consuming and inconvenient.
An apparatus that overcomes the above limitation is disclosed in
U.S. Pat. No. 4,681,450 and in Azzam, R.M.A., et al.,
"Construction, Calibration, and Testing of a Four-Detector
Photopolarimeter," Rev. Sci. Instruments, 59(1), January, 1988,
pages 84-88. This apparatus comprises a series of four
photodetectors serially located in the path of a light beam whose
polarization state is to be measured. The light beam successively
strikes each of the first three photodetectors obliquely, and is
partially specularly reflected. Each specular reflection changes
the state of polarization of the reflected light beam. Each
photodetector produces an electrical signal proportional to the
absorbed portion of the optical energy. The light beam is
substantially fully absorbed in the fourth photodetector. The
electrical signals produced by the four photodetectors can be used
to calculate the Stokes parameters of the incident light beam,
which determines the polarization state of the beam. Since this
apparatus does not involve any mechanical movement, it does not
have the speed limitation of the previously described apparatus or
the inconvenience of a manual measurement.
The apparatus disclosed in U.S. Pat. No. 4,681,450 does, however,
suffer from the drawback that the change in the polarization state
of the light beam reflected at each of the photodetector surfaces
is substantially wavelength-dependent. This apparatus must be
calibrated by using four calibration light beams of different known
polarization states. The calibration must be repeated for each
different wavelength. Furthermore, at least one of the calibration
light beams must not be linearly polarized; and such a beam is
inconvenient to generate accurately. Calibration of the apparatus
disclosed in U.S. Pat. No. 4,681,450 is, therefore, a formidable
task. Consequently, the efficiency and accuracy of the apparatus is
limited, particularly when polarization states of several different
light beams are to be measured.
Another apparatus for performing polarization measurements is
disclosed in U.S. Pat.No. 4,158,506. This apparatus, which is
indicated to be suitable for measuring the polarization state of
nanosecond optical pulses, comprises an assembly of six
photodetectors positioned behind a corresponding assembly of linear
polarizers and waveplates. A light beam passes through all of the
linear polarizers simultaneously, and the transmitted light
intensity from each polarizer is detected and measured by a
corresponding photodetector. The electrical signals produced by the
six photodetectors can then be used to determine the Stokes
parameters of the incident light beam to indicate its polarization
state.
Finally, another optical polarization measurement apparatus is
disclosed in European Patent Application No. 8817382. In this
apparatus, an incident light beam passes through a beam expander,
and four separate portions of the beam pass through four coplanar
Stokes filters. The four portions of the light beam are focused
onto four associated photodetectors that measure the intensities of
the received light. The electrical signals produced by the
photodetectors are used to determine the Stokes parameters of the
incident light beam to indicate its polarization state.
The apparatuses disclosed in U.S. Pat. No. 4,681,450 and European
Patent Application No. 8817382 suffer from the same drawback, in
that there is no provision for ensuring that the incident light
beam whose polarization state is to be measured is properly aligned
relative to the optical elements so that all photodetectors are
subjected to the same beam. Furthermore, there is no provision for
calibrating either apparatus. While European Patent Application No.
8817382 discloses an optical fiber input, and describes the
phenomenon of "polarization noise" that results from transmission
of a light beam through fiber optic material, no technique is
disclosed for correcting for the polarization distortion of the
input fiber.
Additionally, U.S. Pat. No. 4,306,809 describes apparatus having
optical elements that are rotated automatically for determining the
polarizing properties of a material on which a light beam impinges
by measuring the Mueller matrix. However, neither this apparatus,
nor the apparatuses described above, enable the polarization
sensitivity of an optical device to be determined in response to
various polarization states of an incident light beam.
In this regard, accurate characterization of optical devices is
becoming increasingly important as optical devices become more
complex and applications for optical devices proliferate, for
example, in fiber optic telecommunications. One of the fundamental
specifications of any optical device with an optical input and an
optical output is polarization sensitivity, that is, the variation
of optical power transmitted through the device (or reflected by
the device) as the state of polarization incident on the input of
the optical device is varied. For example, the splitting ratio and
excess loss of a fiber optic directional coupler, the insertion
loss of an optical isolator, and the gain of an optical amplifier
all can exhibit variation as the input state of polarization is
altered. In order to use such an optical device effectively in most
practical applications, the polarization sensitivity of its
transmission and/or reflection characteristics must be known.
Conventionally, polarization sensitivity of an optical device under
test (DUT) has been directly measured by monitoring the output
power of the optical DUT with a polarization-independent detector
or optical power meter while the input state of polarization of an
optical source is varied over all possible polarization states.
This is a difficult and time-consuming technique.
Moreover, many arrangements have been devised to transform the
fixed output state of polarization of an optical source into any
desired state of polarization. Such arrangements are generally
referred to as polarization controllers. For example, two
independently rotatable quarter-wave plates constitute a
polarization controller suitable for a light beam propagating
through open space, and two or more single-mode optical fiber loops
of variable orientation can serve as a polarization controller in
fiber optic systems. See, LeFevre, H. C., "Single-mode fibre
fractional wave devices and polarisation controllers," Elect.
Lett., 16, 1980, pages 778-780. Both of these polarization
controllers are manually driven and do not lend themselves to
automation.
Alternatives for polarization controllers exist which can be
electronically controlled. For example, polarization controllers
based on stain-induced birefringence in an optical fiber, which is
effected by piezoelectric or electromagnetic elements, have been
demonstrated, as have polarization controllers based on
electro-optic crystals or waveguides. See, Walker, N. G., and
Walker, G. R., "Polarization control for coherent communications,"
J. Lightwave Technol., 8, 1990, pages 438-458. These polarization
controllers are more easily automated, but an instrument employing
any such polarization controller to measure polarization
sensitivity has two fundamental disadvantages. One disadvantage is
that the control inputs to a polarization controller do not
correlate to the output state of polarization in an easily
ascertainable manner, especially as the wavelength of the optical
source varies. Moreover, the output intensity of the polarization
controller is usually a weak function of the control inputs, and
this variability in intensity translates directly into inaccuracy
in the overall measurement. A second, more serious disadvantage is
the necessity of a search algorithm. The state of a completely (not
partially) polarized optical source has two degrees of freedom, so
it is necessary to vary the state of polarization at the output of
the polarization controller over a two-dimensional space while
searching for the maximum and minimum transmission (or
reflection).
Therefore, a method and apparatus for facilitating determination of
polarization sensitivity of an optical device under test to various
polarization states of an incident light beam are needed so that
the response characteristic of the optical device to different
polarization states can be assessed, for example, during the design
of the optical device. Moreover, such a polarization sensitivity
determination desirably would be calibrated, accurate, and rapidly
obtained, as well as convenient to perform.
SUMMARY OF THE INVENTION
It is an object of this invention to provide an instrument that is
capable of generating different states of polarization of a light
beam and measuring the polarization states produced by the effect
of an optical device under test on the beam to enable polarization
sensitivity of the optical device to be determined.
Another object is to provide an instrument that determines such
polarization sensitivity over a substantial range of
wavelengths.
A further object of the invention is to provide an instrument for
determining polarization sensitivity that is convenient to
calibrate accurately.
Yet another object of the invention is to provide a method for
calibrating an instrument to correct for the distortion of the
polarization state caused by input fiber optics of an optical
polarization meter incorporated into the instrument by using no
more than two reference light beam sources of known
polarization.
Another object of the invention is to provide a method for
calibrating such an instrument to correct for the distortion of the
polarization state caused by any optical network, which is not
completely polarizing, in the input path of an optical polarization
meter incorporated into the instrument by using no more than three
reference light beam sources of known polarization.
One embodiment of the present invention provides a method and
apparatus for determining polarization sensitivity of the
transmission or reflection of an optical device under test using a
polarized optical source, which provides at least three states of
polarization, and an optical polarization meter. The method in
accordance with one embodiment of the invention measures the
responses of an optical device under test to an incident light beam
by providing a light beam having three sequential polarization
states, corresponding to three Jones input vectors, impinging the
beam on the optical device, and splitting the transmitted or
reflected beam into four beams, passing three of the beams through
optical elements, and measuring the intensities of all four beams
by means of photodetectors. The Stokes parameters are then
calculated from the results of these measurements and converted to
Jones output vectors. The Jones matrix for the optical device under
test is then computed to within a complex constant. Thereafter, the
relative polarization sensitivity of the optical device under test
can be determined from this matrix.
Preferably, an optical source is connected to a polarization
synthesizer which is used to sequentially transform the state of
polarization of the light beam generated by the optical source to
three known states of polarization, for example, horizontal,
60-degree, and 120-degree linear polarization states. The three
states of polarization need not be of the same intensity. The light
beam produced by the polarization synthesizer is fed to the optical
device under test having an unknown Jones matrix. The light beam is
impinged on the optical device under test, and the transmitted or
reflected portion of the light beam is routed to the optical
polarization meter which measures the state of polarization. The
optical polarization meter need not measure optical power.
The optical device under test can be situated in an open beam, or
connections can be effected with single-mode optical fiber. The
polarization dependence of the transmission loss of a single-mode
optical fiber is typically small enough that it cannot be measured
and can, therefore, be ignored.
In the optical polarization meter, the received portion of the
light beam is subdivided into four beams and processed by three
sets of optical elements. One of the optical elements is a
horizontal linear polarizer, the second is a linear polarizer with
a polarization direction oriented at a 45-degree angle about the
optical axis relative to the first, and the third element is a
circular polarizer. Measurement of the fourth beam provides a
normalizing factor proportional to the total incident intensity,
that enables determination of all four Stokes parameters. The
received portion of the light beam preferably enters the optical
polarization meter through a single-mode optical fiber that acts as
a spatial filter which, together with other optical elements,
controls the position and alignment of the received portion of the
light beam in the optical polarization meter. An optical fiber
calibration method is provided to correct for polarization
distortion caused by the fiber optic input by using two reference
light sources of known linear polarization.
The three known sequential input states of polarization produced by
the polarization synthesizer yield three Jones input vectors. The
Stokes parameters for the response of the optical device to each of
three sequential polarization states are converted to three
corresponding Jones output vectors. The Jones matrix for the
optical device under test is then computed to within a complex
constant from the Jones input and output vectors. Finally, relative
polarization sensitivity can be determined from this matrix for the
optical device under test. Also, if it is desired to calibrate out
the effect of any optical network which is not completely
polarizing, including the optical device under test, the relative
distortion of the polarization state caused by the optical network
can be corrected by determining the matrix of the optical network
using the three sequential input states of polarization and
multiplying Jones output vectors by the inverse of the matrix of
the optical network during all subsequent measurements through the
optical network. Additionally, power measurements on the optical
device under test and/or optical network, as well as a substituted
optical through (the atmosphere or a single-mode optical fiber),
enable the Jones matrix to be computed so that absolute
polarization sensitivity can be determined and absolute
calibrations can be performed.
BRIEF DESCRIPTION OF THE DRAWINGS
The above and other features of the invention and the concomitant
advantages will be better understood and appreciated by persons
skilled in the field to which the invention pertains in view of the
following description given in conjunction with the accompanying
drawings. In the drawings:
FIG. 1 is a schematic diagram of an instrument for determining the
polarization sensitivity of an optical device under test in
accordance with the invention based on transmission
measurements;
FIG. 2 is a schematic diagram in perspective view of an optical
polarization meter incorporated into the instrument shown in FIG.
1, illustrating the optical path of the received portion of the
light beam from an optical device under test;
FIG. 3 is a flow chart of one embodiment of the method in
accordance with the invention for determining polarization
sensitivity of an optical device under test and for calibrating to
correct for polarization distortion caused by the optical device
under test;
FIGS. 4A and 4B are a flow chart of one embodiment of a method in
accordance with the invention for determining absolute polarization
sensitivity and for absolutely calibrating to correct for
polarization distortion of any input optical network which is not
completely polarizing;
FIG. 5 is a schematic diagram similar to FIG. 1, in which the
instrument is configured for determining the polarization
sensitivity of an optical device under test in accordance with the
invention based on reflection measurements; and
FIG. 6 is a flow chart of one embodiment of a method in accordance
with the invention for calibration to correct for polarization
distortion of the input optical fiber incorporated into the optical
polarization meter shown in FIG. 2.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
One embodiment of an instrument for achieving calibrated, accurate,
convenient polarization sensitivity determinations in accordance
with the invention, generally indicated by the numeral 8, is shown
in FIG. 1. The instrument 8 comprises a polarized optical source 9
for feeding a light beam l to an optical device under test (DUT)
30, and an optical polarization meter 10 for receiving a portion of
the light beam .DELTA.l transmitted by the optical DUT. Actually,
the polarized optical source 9 can comprise an optical source 40
and a polarization synthesizer 50, as indicated by the dashed lines
surrounding these elements in FIG. 1. For example, the optical
source 40 can be a solid-state laser which generates a light beam
at a given wavelength, such as 1300 nanometers. The polarization
synthesizer 50 is preferably automated to sequentially insert three
different polarizers 50a, 50b, and 50c into the path of the light
beam generated by the optical source 40 to produce three sequential
states of polarization of the light beam l. Conveniently, the
polarizers 50a, 50b, and 50c preferably transmit linear
polarization states, but, alternatively, they can be selected to
transmit elliptical states of polarization. Other polarization
generators can also be used, including a variable or rotatable
waveplate.
As shown in FIG. 2, the portion of the light beam .DELTA.l
transmitted by the optical DUT 30 enters the optical polarization
meter 10 through a single-mode optical fiber 11, which provides
spatial filtering of the beam. A method for calibrating to correct
for the distortion of the polarization state caused by the input
optical fiber 11 by using no more than two reference light beams of
known polarization will be described later.
The optical polarization meter 10 can operate in the range of
wavelengths over which the input optical fiber 11 supports a single
mode of propagation. For example, a standard long-haul
telecommunications fiber manufactured by Corning Glass for
1.3-micron transmission can support a single propagation mode over
the wavelength range of 1.2-1.6 micrometers.
The light beam .DELTA.l proceeds into a focusing concave mirror 12
sectioned into independently movable quadrants 12a, 12b, 12c, and
12d. The efficiency and accuracy of the optical polarization meter
10 are enhanced by the spatial filtering function of the input
optical fiber 11 which ensures that the light beam .DELTA.l is
repeatably distributed among the quadrants 12a-12d. The focusing
concave mirror 12 splits the beam into four separate beams, as
shown in FIG. 2. The four beams are sufficiently separated to allow
different optical elements to be placed in their path. The beams
are denoted by ".DELTA.la," ".DELTA.lb," ".DELTA.lc," and
".DELTA.ld,"
The beam labelled ".DELTA.lb" passes through a linear polarizer 14
having a horizontal polarization axis. Beam ".DELTA.lc" passes
through a linear polarizer 15 having a polarization axis oriented
at a 45-degree angle about the optical axis relative to the linear
polarizer 14. Beam ".DELTA.ld" passes through a quarter-wave plate
17, and then through a linear polarizer 16 that is oriented in the
same direction as the linear polarizer 15; this combination is a
circular polarizer. Finally, beam ".DELTA.la" has no optical
elements in its path.
The four beams ".DELTA.la"-".DELTA.ld" impinge on a respective
photodetector 18, 19, 20, or 21 and are substantially absorbed.
Each photodetector 18-21 produces an electrical signal that is
proportional to the intensity of the optical energy absorbed. The
photodetector 18 produces a signal of amplitude T, photodetector 19
produces a signal H, photodetector 20 produces a signal F, and
photodetector 21 produces a signal C. The measurement frequency of
the optical polarization meter 10 is limited only by the frequency
response of the photodetectors 18-21, which can easily exceed 1
GHz.
Each quadrant 12a-12d of the focusing concave mirror 12 is adjusted
to focus the portion of the light beam .DELTA.l impinging on it
onto the corresponding photodetector 18-21 either directly or
through the optical elements 14-17, as the case may be. The
adjustment mechanism for the quadrant mirror sections 12a-12d is
not shown, since it can be constructed readily by persons of skill
in the field of the invention.
The electrical signals produced by the photodetectors 18-21 are
routed to a microprocessor 27 having an analog-to-digital converter
circuit. The amplitude of the electrical signals produced by the
photodetectors 18-21 can be used to determine the Stokes parameters
of the portion of the light beam .DELTA.l transmitted by the
optical DUT 30 shown in FIG. 1. For purposes of this description,
definition of the Stokes parameters is based on the well-known
treatise entitled Principles of Optics, by M. Born and E. Wolf
(Pergamon Press, 4th Edition, London, 1970, pages 30-32). These
parameters are denoted by the symbols "s.sub.0," "s.sub.1,"
"s.sub.2," and "s.sub.3," and specification of all four of these
quantities, based on the known configuration of the optical
elements 14-17 and the intensities measured by the photodetectors
18-21, determines the state of polarization of the light beam
l.DELTA.. The electrical signals T, H, F, and C produced by the
photodetectors 18-21 are related to the Stokes parameters by the
expressions:
The Stokes parameter s.sub.0 is simply the total light intensity.
The Stokes parameters s.sub.1, s.sub.2, and s.sub.3 are determined
from the electrical signals produced by the photodetectors 18-21 by
Equations (2)-(4). The degree of polarization is given by the
expression: ##EQU1## Such calculations can be carried out
automatically by the microprocessor 27 shown in FIG. 2.
Previously, to measure polarization sensitivity of an optical
device under test (DUT), the polarization sensitivity of the
optical DUT has been directly measured by monitoring the output
power of the optical DUT with a polarization-independent detector
or optical power meter, while the input state of polarization of an
optical source is varied over all possible polarization states. The
accuracy and convenience of the conventional polarization
sensitivity measurement have, however, been less than
desirable.
For the purposes of the following description, determination of
polarization sensitivity of the optical DUT 30 shown in FIG. 1 will
be described for the case entailing measurements of transmission of
the light beam l through the optical DUT. Alternatively,
measurements of reflection can be performed instead of transmission
measurements by employing a beam splitter or directional coupler,
as will be briefly described later. Determination of polarization
sensitivity based on either transmission or reflection measurements
can be understood by analyzing the case of transmission, which will
now be described.
A suitable description of polarization sensitivity depends upon the
topology of the optical DUT. The polarization sensitivity of a
simple optical network having one optical input and one optical
output can be characterized by a single real number which expresses
the ratio of the maximum to the minimum intensity transmission
coefficient as the input state of polarization of the incident
light beam is varied over all possible polarization states.
Similarly, an optical network having n inputs and m outputs can be
characterized by an n.times.m matrix of real numbers, each number
denoting the polarization sensitivity of a particular input-output
pair. The method for determining polarization sensitivity in
accordance with the invention applies to any optical network.
A useful, compact formalism for the treatment of polarization
characteristics in optical systems was introduced by R. C. Jones
during the years 1941-1956. See, Jones, R. C., "A new calculus for
the treatment of optical systems. I. Description and discussion of
the calculus," J. Optical Soc. Am., 31, 1941, pages 488-493; "A new
calculus for the treatment of optical systems. II. Proof of three
general equivalence theorems," J. Optical Soc. Am., 31, 1941, pages
493-499; "A new calculus for the treatment of optical systems. III.
The Sohncke theory of optical activity," J. Optical Soc. Am., 31,
1941, pages 500-503; "A new calculus for the treatment of optical
systems. IV.," J. Optical Soc. Am., 32, 1942, pages 486-493; "A new
calculus for the treatment of optical systems. V. A more general
formulation and description of another calculus," J. Optical Soc.
Am., 37, 1947, pages 107-110; "A new calculus for the treatment of
optical systems. VI. Experimental determination of the matrix," J.
Optical Soc. Am., 37, 1947, pages 110-112; "A new calculus for the
treatment of optical systems. VII. Properties of the N-matrices,"
J. Optical Soc. Am., 38, 1948, pages 671-685; "A new calculus for
the treatment of optical systems. VIII. Electromagnetic theory," J.
Optical Soc. Am., 46, 1956, pages 126-131. A synopsis of the Jones
calculus is presented in Chapter 4 of Kliger, D. S., Lewis, J. W.,
and Randall, C. E., Polarized light in optics and spectroscopy,
Academic Press, San Diego, 1990.
Generally, Jones derived an explicit expression for experimentally
determining the forward transmission matrix M of an unknown,
linear, time-invariant optical device (Jones, R. C., "A new
calculus for the treatment of optical systems. VI. Experimental
determination of the matrix," J. Optical Soc. Am., 37, 1947, pages
110-112). This restriction precludes optical devices that generate
new optical frequencies different from those of the incident light
beam.
Also, a Jones vector cannot be employed to represent a partially
polarized field. However, this is not a practical limitation, since
a light beam from an optical source can be completely polarized by
a linear polarizer.
Furthermore, a Jones matrix cannot represent a depolarizing optical
DUT. However, depolarizing effects can be eliminated by using a
quasi-monochromatic optical source with a sufficiently long
coherence length.
Subject to these constraints, the Jones calculus can express an
input optical electric field by a one-by-two complex column field
vector which completely specifies the phase and state of
polarization of a light beam, such as the light beam l shown in
FIG. 1. The two complex elements of this vector represent the
amplitude and phase of the x and y components of the optical
electric field, respectively. An optical DUT, such as the optical
DUT 30, is represented by a complex two-by-two matrix. The effect
of an optical DUT on an input optical electric field is found by
multiplying the field vector by the matrix representing the optical
DUT to obtain an output optical electric field vector which
represents the transmitted portion of the light beam .DELTA.l.
By way of contrast, one embodiment of the method in accordance with
the invention for determining polarization sensitivity of the
optical DUT 30 is based on specification of the input optical
electric field Jones vectors for three known states of polarization
of the light beam l, performing intensity measurements on the
transmitted portion of the light beam .DELTA.l needed to derive the
output optical electric field Jones vectors, and computing the
Jones matrix for the optical DUT 30 from the input and output Jones
vectors. Thereafter, the method of the invention determines the
polarization sensitivity of the optical DUT 30 from the computed
Jones matrix. This provides an accurate determination of the
polarization sensitivity of the optical DUT 30, as well as
minimizes the number of actual measurements that must be performed,
and, therefore, is rapid and convenient. Advantageously, this
method also enables calibration to correct for the distortion of
the polarization state caused by any optical network, which is not
completely polarizing, in the input path of the optical
polarization meter 10 by using no more than three reference light
beams of known polarization, as will be described later. First,
however, the polarization sensitivity determination method of the
invention will be described in more detail.
One embodiment of the method in accordance with the invention
determines the polarization sensitivity of the transmission of the
optical DUT 30 shown in FIG. 1 in response to three sequential
known states of polarization of the light beam l. The output of the
optical source 40 is connected to the input of the polarization
synthesizer 50 which is used to sequentially transform the state of
polarization of the beam of light generated by the optical source
40 to three sequential predetermined polarization states, for
example, horizontal, 60-degree, and 120-degree linear polarization.
The three states of polarization need not be of the same
intensity.
The output of the polarization synthesizer 50 is connected to the
input of the optical DUT 30. The optical DUT 30 has an unknown
transmission Jones matrix. As will be shown, polarization
sensitivity of the optical DUT 30 can be determined from this Jones
matrix once the matrix is determined. In accordance with the method
of the invention, the unknown Jones matrix is computed from the
measured responses (Stokes parameter measurements) of the optical
DUT 30 to the three sequential predetermined polarization states of
the light beam l.
Considered in more detail, one embodiment of the method for
determining the polarization sensitivity of the optical DUT 30 is
shown in FIG. 3. Polarization sensitivity can be defined to be the
ratio of the maximum transmission for any state of polarization to
the minimum transmission for any polarization state through the
optical DUT 30. Generally, the polarization sensitivity for the
case of transmission can be expressed as t.sub.max /t.sub.min.
As indicated above, the one embodiment of the method in accordance
with the invention for determining polarization sensitivity of the
optical DUT 30 employs Jones calculus. Use of Jones calculus to
determine the polarization sensitivity of the optical DUT 30
requires that the light beam l fed to the optical DUT be of a known
state of polarization. Nevertheless, the optical source 40 can be
any source of optical energy, and the beam of light generated by
the optical source can have any polarization, including a beam
which is not linearly polarized, such as an elliptically polarized
beam or an unpolarized beam. This is because the polarization
synthesizer 50 assures that the light beam generated by the optical
source 40 is of a known polarization state when the light beam l
exits the polarization synthesizer. If, however, the optical source
40 generates a linearly polarized light beam, selection of the
optical source and the sequential settings of the polarization
synthesizer 50 must be such that the polarization state of the beam
generated by the optical source does not result in complete
filtering (i.e., cancellation) of the beam by one of the settings
of the polarization synthesizer.
In accordance with the one embodiment of the method of the
invention for determining polarization sensitivity of the optical
DUT 30, the polarization synthesizer 50 is sequentially set to
three predetermined polarization settings so that the light beam l
fed to the optical DUT has three predetermined polarization states,
as indicated by the numeral 60 shown in FIG. 3. For example, the
polarization synthesizer 50 can be set to sequentially produce
linear polarization states at 0.degree., 60.degree., and
120.degree.. Because the polarization states are known, and because
these polarization states are linear, three input optical electric
field Jones vectors can be specified, as indicated by the step 60
shown in FIG. 3, namely: ##EQU2## where i=1, 2, 3 corresponding to
the three sequential settings of the polarization synthesizer 50,
and .theta..sub.i is the angle corresponding to the polarization
state at the present setting.
The light beam l having the three sequential predetermined input
states of polarization is fed either through the atmosphere (open
beam) or through the optical fiber 51 to the optical DUT 30, as
indicated by the numeral 62 shown in FIG. 3. The optical DUT 30
affects the polarization state of the light beam produced by each
of the three sequential predetermined settings of the polarization
synthesizer 50.
As mentioned above, polarization sensitivity of the optical DUT 30
cannot be accurately determined using Jones calculus if the optical
DUT is depolarizing. However, the effects of a depolarizing optical
DUT 30 can be eliminated by using an optical source 40 having a
very narrow spectral line width, that is, by using an optical
source which is quasi-monochromatic with a sufficiently long
coherence length.
The polarization sensitivity of the optical DUT 30 can be
determined from the three sequential known input electric field
Jones vectors given by Equation (6) and the three measured
responses of the optical DUT to the three sequential predetermined
input states of polarization, as follows. In the case of
transmission through the optical DUT 30, the respective
polarization states of the light beam l produced by the
polarization synthesizer 50 sequentially impinge on the optical
DUT. The three resulting polarization states of the portion of the
light beam .DELTA.l sequentially exit the optical DUT 30 and are
fed either open beam or by the optical fiber 52 to the optical
polarization meter 10. As indicated by the numeral 64 shown in FIG.
3, the optical polarization meter 10 measures the Stokes
parameters, as described earlier, from which the polarization state
of each of the three sequential polarization states of the light
beam .DELTA.l produced by the optical DUT 30 can be calculated.
The optical polarization meter 10 measures the Stokes parameters of
the three sequential polarization states of the light beam .DELTA.l
received by the optical polarization meter and computes the
corresponding Stokes vectors, as indicated by the numeral 66 shown
in FIG. 3. This is accomplished by measuring the signals H, T, C,
and F shown in FIG. 2 for each received polarization state of the
light beam .DELTA.l. The Stokes vectors for the three sequential
predetermined input polarization states are: ##EQU3## where i=1, 2,
3 and corresponds to the three sequential states of polarization of
the light beam .DELTA.l due to the three sequential predetermined
settings of the polarization synthesizer 50, the matrix is the
instrument matrix of the optical polarization meter 10 specified by
the presence of the optical elements 14-17, and H, T, C, and F are
the electrical signals produced by the photodetectors 18-21. In
practice, the instrument matrix may vary from that shown in
Equation (7) as a result of imperfections in the optical elements
14-17 and uneven distribution of the light beam .DELTA.l among the
photodetectors 18-21 and, therefore, must be determined.
Jones calculus can be employed to determine the polarization
sensitivity of the optical DUT 30 only if relative phase
information between the x and y components of the optical electric
field are present. The method of the invention for determining
polarization sensitivity recognizes that the Stokes vectors s.sub.i
contain relative phase information between the x and y components
of the optical electric field. Therefore, the Stokes vectors can be
converted to output electric field Jones vectors, as indicated by
the numeral 68 shown in FIG. 3, as will now be described.
Initially, the degree of polarization given by Equation (5) of each
of the three Stokes vectors is set equal to one by changing the
measured value of s.sub.i0 to s'.sub.i0 : ##EQU4## where i=1, 2, 3.
This reduces the effect of noise.
Also, the following normalized Stokes parameters needed for
conversion of the Stokes vectors to output electric field Jones
vectors are defined: ##EQU5## The output electric field Jones
vectors can now be derived from the Stokes vectors for each of the
three sequential settings of the polarization synthesizer 50 in
accordance with the following conversion expression based on a
normalized Stokes vector having unity degree of polarization:
##EQU6##
Now, the Jones matrix of the optical DUT 30 is defined as: ##EQU7##
Therefore, ##EQU8## A is the Jones matrix M to within a complex
constant k. Elements a, b, and c of the matrix A can be computed
from the three measured responses represented by Jones output
vectors r.sub.i and the three sequential predetermined input
polarization states specified by Jones input vectors q.sub.i.
Let: ##EQU9##
The magnitudes and absolute phases of the Jones vectors r.sub.i and
q.sub.i are not required for this calculation, so the above
expression can be simplified by converting Jones vectors to complex
numbers, first by mapping Jones vectors onto the Poincare sphere,
and then by mapping the Poincare sphere onto the complex plane. The
mapping of Jones vectors and Stokes vectors onto the Poincare
sphere is accomplished simply by defining the normalized Stokes
parameters given by Equation (9) to be the x, y, and z Cartesian
coordinates of points in space. See chapter 1 of Born, M., and
Wolf, E., Principles of Optics, 6th Edition, Pergamon Press, New
York, 1989. This establishes a one-to-one mapping between points on
the unit sphere and Jones vectors of unity optical power. The
mapping from the Poincare sphere onto the complex plane is a
stereographic projection. See, Churchill, R. V., Brown, J. W., and
Verhey, R. E., Complex variables and applications, 3rd Edition,
McGraw-Hill, New York, 1976, page 20.
Suppose the Poincare sphere is centered at the origin of a
horizontal complex plane. The sphere can be defined to have a pole
located directly above the origin. A point P on the sphere is
mapped onto the plane by a line which intersects the pole and the
point P on the sphere. The line intersects the complex plane at the
point P', and P' is known as the projection of P. When the Poincare
sphere is oriented so that the point representing horizontal linear
polarization is located at the pole and the point representing
+45-degree linear polarization is located at the point (1,0) on the
complex plane, the projection P' is given by the ratio P.sub.x
/P.sub.y (the ratio of the x and y components of the Jones vector).
Writing Equation (13) in terms of the projections r'=r.sub.x
/r.sub.y and q'=q.sub.x /q.sub.y yields: ##EQU10## Note that r' and
q' are complex scalars, whereas r and q are complex vectors.
Consequently, after measuring three responses to three sequential
predetermined input states of polarization, a matrix Z can be
defined: ##EQU11## From this last expression, the desired matrix A
can be computed, as indicated by the numeral 70 shown in FIG. 3, by
measuring r'.sub.1, r'.sub.2, r'.sub.3 and the elements of Z, then
by inverting Z and performing the indicated matrix
multiplication.
The mathematics can be simplified by selecting a stereographic
projection in which the point q'.sub.i on the Poincare sphere is
located directly below the origin of the complex plane
diametrically opposite the pole of the sphere. In that case,
q'.sub.1 =0, and, therefore, r'.sub.1 =b from Equation (14).
Consequently, the above three-by-three complex linear system of
equations can be reduced to a two-by-two complex system:
Subtracting r'.sub.1 from both sides of Equation (17) and rewriting
the equation in matrix form, the following expression is obtained:
##EQU12## Therefore, ##EQU13## Hence, b=r'.sub.1, and a and c can
be computed from Equation (19), which specifies the matrix A, as
indicated by the step 70 shown in FIG. 3. Y is invertible if det
(Y).noteq.0, i.e., if q'.sub.2 .noteq.0 and q'.sub.3 .noteq.0 and
r'.sub.2 .noteq.r'.sub.3. If the matrix A is invertible (i.e., does
not represent a perfect polarizer), then q.sub.i =k.sup.-1 A.sup.-1
r.sub.i, and r'.sub.2 .noteq.r'.sub.3 implies q'.sub.2
.noteq.q'.sub.3. Remembering that q'.sub.1 =0, Y is invertible if
the matrix A is invertible and q'.sub.2 and q'.sub.3 differ from
q'.sub.1 and differ from each other.
Also, considering that measurements are performed in the presence
of noise and small systematic errors, the most accurate calculation
of the matrix A will result when the three sequential predetermined
input states of polarization are as far as possible apart from one
another on the Poincare sphere. This leads to the following
preferred selection of the three sequential input states of
polarization.
Since it is easier to generate linear states of polarization at any
wavelength than it is to generate elliptical polarization states,
the input states of polarization are preferably generated by
successively inserting three linear polarizers into a light beam
which is approximately unpolarized or approximately circularly
polarized. This constrains the powers of the three sequential input
states of polarization to be roughly equal and yields most accurate
measurement. Accordingly, the linear polarizers 50a, 50b, and 50c
at relative angles 0.degree., 60.degree., and 120.degree. are
preferably selected so that the three sequential predetermined
input states of polarization specified by the Jones input vectors
q.sub.i will be located at 120.degree. intervals on a great circle
on the Poincare sphere, i.e., as far apart as possible.
A few techniques from linear algebra needed to understand the
determination of polarization sensitivity of the optical DUT 30
from the matrix A in accordance with one embodiment of the method
of the invention will now be briefly described. Notation follows
that of Lancaster, P., and Tismenetsky, M., The theory of matrices,
2nd Edition, Academic Press, San Diego, 1985.
A standard inner product (x, y)=y* x can be defined to associate a
scalar with any pair of complex vectors x and y. (y* denotes the
conjugate transpose of y, and c denotes the complex conjugate of a
complex scalar c.) The intensity of an optical electric field
represented by a Jones vector x is proportional to the inner
product (x, x), as is mentioned in Wanser, K. H., and Sabar, N. H.,
"Remote polarization control for fiber-optic interferometers,"
Optics Lett., 12, 1987, pages 217-219.
Additionally, the field of values of a matrix L is defined as the
set of complex numbers (Lx, x), where x ranges over all vectors
that are normalized so that (x, x)=x* x=1. It can be shown that the
field of values of a Hermitian matrix is an interval of the real
line, and that the eigenvalues of a Hermitian matrix are real.
Furthermore, the maximum and minimum of the field of values of a
two-by-two Hermitian matrix with eigenvalues .lambda..sub.1 and
.lambda..sub.2, with .lambda..sub.1 .ltoreq..lambda..sub.2, are
given by those eigenvalues, i.e., the minimum for the field of
values is .lambda..sub.1 and the maximum is .lambda..sub.2.
Furthermore, the matrix product B* B is positive semi-definite, so
the square root of the product (B* B).sup.1/2 exists. The singular
values .sigma..sub.i of B are the eigenvalues of this square root,
i.e., .sigma..sub.i (B)=.lambda..sub.i (B* B).sup.1/2). Singular
values are non-negative real numbers. It can be shown that
.sigma..sub.i.sup.2 (B)=.lambda..sub.i (B* B), and that the
singular values of a square matrix are invariant under unitary
transformation, i.e., for any square matrix B and any two unitary
matrices C and D, .sigma..sub.i (B)=.sigma..sub.i (CBD).
Now, to determine polarization sensitivity of the optical DUT 30,
it is desired to find maximum and minimum intensity transmission
coefficients t.sub.max and t.sub.min through the optical DUT over
the range of all possible states of polarization. The input optical
electric field to the optical DUT 30 is given by the Jones input
vector q, the output optical electric field is Aq, so the problem
is reduced to finding the maximum and minimum values of (Aq, Aq)
over all inputs q of a constant intensity (q, q). Using linear
algebra, it can be shown that (Aq, Aq)=(Pq, q), where P=A* A is
Hermitian. Finding the maximum and minimum intensity transmission
coefficients is therefore equivalent to finding the maximum and
minimum of the field of values of P, which are given by
.lambda..sub.1 (P) and .lambda..sub.2 (P), which are the
eigenvalues of P. Consequently, having determined the matrix A at
the step 70 shown in FIG. 3, the polarization sensitivity (or
dependence) of the power transmission through the optical DUT 30
can be determined, as indicated by the numeral 72 shown in FIG. 3,
as follows.
The expression for polarization sensitivity obtained in terms of
the singular values of the matrix A for global variation in power
transmission through the optical DUT 30 is given by: ##EQU14##
where .sigma..sub.1 (A) and .sigma..sub.2 (A) are the singular
values of A. .sigma..sub.i.sup.2 (A)=.lambda..sub.i (A* A), where
i=1,2. .lambda..sub.i (A* A) are the eigenvalues of A* A, and A* is
the conjugate transpose of A. This leads to the following
expression for the determination of polarization sensitivity:
##EQU15## and d=2Re[abc], where a is the complex conjugate of a.
When the three sequential predetermined input states of
polarization are at the relative angles 0.degree., 60.degree., and
120.degree., the projected stimuli are q'.sub.1 =0, (b=r'.sub.1),
q'.sub.2 =tan 60.degree.=.sqroot.3, and q'.sub.3 =tan
120.degree.=-.sqroot.3.
If optical fibers are present, such as the optical fibers 51 and
52, then the responses of these fibers are included in the
responses of the optical DUT 30. However, the polarization
dependence of the transmission loss of a single-mode optical fiber
is small enough that it typically cannot be measured, and this
leads to a Jones matrix which is unitary. See, Wanser, K. H., and
Sabar, N. H., "Remote polarization control for fiber-optic
interferometers," Optics Lett, 12, 1987, pages 217-219, and Poole,
C. D., and Wagner, R. E., "Phenomenological approach to
polarisation dispersion in long single-mode fibers," Elec. Lett.,
22, 1986, pages 1029-1030. The optical fibers 51 and 52 connected
to the input and output of the optical DUT 30, respectively, are
therefore labeled with the unitary Jones matrices U and V,
respectively, as shown in FIG. 1, and the transmission matrix of
the optical DUT with optical fiber connections is therefore
VAU.
The polarization sensitivity of the optical DUT 30 given by
Equation (20) is in relative terms t.sub.max /t.sub.min, since the
polarization sensitivity determination is based on the matrix
A=k.sup.-1 M, where M is the actual Jones matrix for the optical
device. The value of the complex constant k can be found so that
the absolute power transmission loss or gain can be determined, as
follows.
The above-described polarization sensitivity determination yields a
ratio in terms of the variation in power loss or gain through the
optical DUT 30 over all states of polarization. By adding a through
calibration, a range of absolute loss or gain can be
determined.
Initially, a through (single-mode optical fiber in the
fiber-coupled case or the atmosphere in the open-beam case), which
is assumed to have negligible loss, is connected between the
polarized optical source 9 and the optical polarization meter 10
instead of the optical DUT 30, as indicated by the numeral 80 shown
in FIG. 4. Then, the optical power I.sub.Fn transmitted by the
through to the optical polarization meter 10 is measured as the
polarization synthesizer 50 produces the three sequential
predetermined input states of polarization (n=1, 2, 3), as
indicated by the numerals 82, 84, and 86 shown in FIG. 4. This
completes the through calibration.
The optical DUT 30 is then substituted for the through, as
indicated by the numeral 88 shown in FIG. 4. Next, the steps 60,
62, and 64 described in connection with FIG. 3 are performed, as
indicated by the primed numerals 60', 62', and 64' shown in FIG. 4.
Additionally, the optical power I.sub.Dn is measured for each of
the same three sequential predetermined input states of
polarization (n=1, 2, 3), as indicated by the numeral 90 shown in
FIG. 4. Thereafter, the steps 66, 68, and 70 described in
connection with FIG. 3 are performed, as indicated by the primed
numerals 66', 68', and 70' shown in FIG. 4. This results in three
optical power measurements for the through, three optical power
measurements for the optical DUT 30, and the matrix A.
Since the through has negligible loss (i.e., a unitary Jones
matrix), from Equation (12): ##EQU16## q.sub.n is related to its
projection q'.sub.n by: ##EQU17## Consequently, ##EQU18## from
which the complex constant k can be computed, as indicated by the
numeral 92 shown in FIG. 4.
Three values of .vertline.k.vertline..sub.n.sup.2 will be computed
(n=1, 2, 3). All three should be equal, but in the presence of
errors and noise, a mean value .vertline.k.vertline..sub.mean.sup.2
=1/3(.vertline.k.vertline..sub.1.sup.2
+.vertline.k.vertline..sub.2.sup.2
+.vertline.k.vertline..sub.3.sup.2) is preferably used for
.vertline.k.vertline..sub.n.sup.2.
Now, the maximum and minimum transmission through the optical DUT
30 will occur at the states of polarization given by the
eigenvectors e.sub.1 and e.sub.2 of A*A or M*M, since the
eigenvectors of these matrix products are identical. If two input
states of polarization are deemed to be the unit amplitude
eigenvectors, the responses of the optical DUT 30 will be r.sub.1
=kAe.sub.1 and r.sub.2 =kAe.sub.2, where k is equal to the square
root of .vertline.k.vertline..sub.mean.sup.2. Hence, ##EQU19## Here
.lambda..sub.1 and .lambda..sub.2 are the eigenvalues of A*A, that
is, .lambda..sub.m e.sub.m =A*Ae.sub.m for m=1, 2. Therefore, as
indicated by the numeral 94 shown in FIG. 4, the absolute range of
power loss or gain is given by:
Similarly, polarization sensitivity can be defined in the case of
reflection to be the ratio of the maximum reflection for any
polarization state to the minimum reflection for any state of
polarization from the optical DUT 30. The polarization sensitivity
for the case of reflection can be generally expressed in relative
terms as r.sub.max /r.sub.min or in absolute terms as R.sub.min and
R.sub.max.
r.sub.max /r.sub.min and R.sub.min and R.sub.max can be determined
analogously to t.sub.max /t.sub.min and T.sub.min and T.sub.max,
respectively, based on reflection measurements instead of
transmission measurements using the instrument shown in FIG. 5.
Elements 9', 10', 30', 51', and 52' shown in FIG. 5 correspond to
the elements 9, 10, 30, 51, and 52, respectively, shown in FIG. 1.
The only difference is that the light beam l is fed to the optical
DUT 30' by a directional optical coupler 100, and the portion of
the beam .DELTA.l fed to the optical polarization meter 10' is a
reflected beam, rather than a beam transmitted by the optical DUT
30 to the optical polarization meter 10, as shown in FIG. 1.
Consequently, both transmission and reflection measurements are
considered to be within the scope of the method for determining
polarization sensitivity in accordance with the invention.
In summary, the polarization sensitive of the optical DUT 30 or 30'
can be determined in relative terms, as shown in FIG. 3, or in
absolute terms, as shown in FIG. 4. In each case, the matrix A is
determined. Since the matrix A can be determined, the principles of
the method in accordance with the invention can be generalized to
correct for the distortion of the polarization state caused by any
optical network which is not completely polarizing, such as the
optical DUT 30 or 30', by determining the matrix A of the optical
network using the three sequential predetermined input states of
polarization and multiplying responses during subsequent
measurements through the optical network represented by Jones
output vectors by the inverse of the matrix A. Additionally, in the
case of the method shown in FIG. 4, this calibration is in absolute
terms.
Considered in more detail, the optical network to be calibrated out
is considered to be the optical DUT for the purposes of the
processes shown in FIGS. 3 and 4. Then, rather than proceed to
determination of the relative polarization sensitivity at the step
72 shown in FIG. 3 or the absolute polarization sensitivity at the
step 94 shown in FIG. 4, the method in accordance with the
invention performs a calibration, as follows.
One embodiment of the method in accordance with the invention also
provides calibration of the instrument shown in FIGS. 1 and 5 to
correct for the distortion of the polarization state caused by any
optical network, which is not completely polarizing, in the input
path of the optical polarization meter 10, using no more than three
reference light beam sources of known polarization. Preferably, the
three sequential predetermined input states of polarization are fed
to the optical network which is desired to be calibrated out of the
optical path between the polarized optical source 9 or 9' and the
polarization meter 10 or 10' to determine the elements a', b', and
c' of a matrix A' by the procedures described above in connection
with FIGS. 3 and 4 for determining the matrix A for an optical
device under test. That is, the optical network to be calibrated
out is effectively considered to be an optical device under test
for the purposes of the preceding description insofar as
determining the matrix A is concerned. Therefore, the matrix A'
from a defined calibration reference frame (for example, the plane
of the input to the optical polarization meter 10) is then:
##EQU20## as indicated by the numeral 102 shown in FIG. 3. Then,
any Jones output vector r measured using the optical polarization
meter 10 can be transformed to the defined calibration reference
frame by multiplying the total response r by the inverse of A'
(i.e., A'.sup.-1), as indicated by the numeral 104 shown in FIG. 3.
Similarly, the Jones matrix M' from a defined calibration reference
frame is ##EQU21## as indicated by the numeral 106 shown in FIG. 4.
Then, any Jones output vector r measured using the optical
polarization meter 10 can be transformed to the defined calibration
reference frame by multiplying the total response r by the inverse
of M' (i.e., M'.sup.-1), as indicated by the numeral 108 shown in
FIG. 4. If the calibrated Jones output vector is v.sub.cal, then
v.sub.cal =M'.sup.-1 v.
Also, any measured Stokes vector s can be transformed to a
calibrated Stokes vector s.sub.cal by multiplying by M.sub.1, the
Mueller matrix equivalent of M'.sup.-1, i.e., s.sub.cal =M.sub.1 s.
M.sub.1 can be derived from M'.sup.-1 in accordance with the
procedure described in Hauge, P. S., et al., Surface Science, 96,
1980, pages 101-107.
If the optical network or optical DUT to be calibrated out is
completely polarizing (behaves as a perfect polarizer), its Jones
matrix M' will be singular, as will be the Mueller matrix
equivalent. Such an optical network or optical DUT cannot be
calibrated out, since the inverse of a singular matrix does not
exist.
Finally, beam alignment and positioning problems inherent in the
apparatuses disclosed in U.S. Pat. No. 4,681,450 and 4,158,506 are
solved in accordance with one embodiment of the method in
accordance with the invention by using the input optical fiber 11
to introduce the light beam .DELTA.l into the optical polarization
meter 10. Such an optical fiber acts as a spatial filter, so that
the direction and distribution of the incoming light beam .DELTA.l
is highly repeatable. However, it is known that such optical fibers
may become birefringent under mechanical strain, such as bending,
and thereby affect the state of polarization of the incident light
beam and the accuracy of polarization measurements. Therefore, a
calibration is also preferably performed to correct for any
polarization effects induced by the input optical fiber 11, as
follows.
The polarization transformation induced by the input optical fiber
11 can be expressed as a real four-by-four Mueller matrix denoted
by the symbol [M]. A short length of fiber has negligible loss and
will not change the degree of polarization of the light beam
.DELTA.l, and, hence, the matrix [M] can be written in terms of an
orthogonal three-by-three sub-matrix [T] as follows: ##EQU22##
The portion of the light beam .DELTA.l described by the Stokes
vector [p], where [p] is a column matrix, will be transformed into
an output beam of polarization [p'] by the input optical fiber 11.
The transformation can be written as a matrix product:
Knowledge of all the elements of the matrix [M] will determine the
correction to be performed. The matrices [M] and [T] have inverses
denoted by [M.sup.-1 ] and [T.sup.-1 ], respectively. It can be
shown that: ##EQU23##
One embodiment of the method in accordance with the invention for
correcting the distortion of the polarization state caused by the
input optical fiber 11 determines the matrix [T], and, therefore,
the matrix [M], by successively introducing two different light
beams of linearly polarized light and measuring the resulting two
polarization states at the output of the input optical fiber 11
using the optical polarization meter 10, as indicated by the
numerals 120 and 122 shown in FIG. 6. The polarization direction of
these two input light beams are preferably at an angle of
45.degree. relative to each other, although any relative angle
between, but not including, 0.degree. and 90.degree. can be used,
including two of the three sequential predetermined input
polarization states described earlier. As indicated by the numeral
124 shown in FIG. 6, a three-element normalized Stokes vector n can
be derived from a full four-element Stokes vector s by dividing the
second, third, and fourth elements of the Stokes vector by the
first element: ##EQU24##
If it is assumed that the first light beam is horizontally
polarized, then the resulting polarization state emerging from the
input optical fiber 11 can be denoted by the normalized Stokes
vector h. The polarization resulting from the second light beam can
be denoted by the normalized Stokes vector f. Three orthonormal
vectors x, y, and z are then formed as follows: ##EQU25##
Any two reference light beams can be used to determine [T], as long
as they correspond to Stokes vectors h and f that have a non-zero
cross product h.times.f, which is true for all relative angles
between their direction of polarization except for 0.degree. and
90.degree..
The elements of x, y, and z are the columns of the desired matrix
[T]: ##EQU26## The desired state of polarization at the input of
the input optical fiber 11 is obtained by forming the inverse of
the matrix [T] and substituting the inverse (i.e., [T.sup.-1 ])
into the matrix [M.sup.-1 ] in Equation (32), as indicated by the
numeral 126 shown in FIG. 6. The inverse Mueller matrix [M.sup.-1 ]
is then used to multiply the measured state of polarization at the
output of the input optical fiber 11 to correct for its distortion
of the polarization state of the light beam .DELTA.l:
as indicated by the numeral 128 shown in FIG. 6.
The computation of this calibration correction of the optical
polarization meter 10 at different wavelengths can be performed by
the microprocessor 27 shown in FIG. 2. A similar calibration can be
carried out not only for optical fibers, but for any birefringent
medium which transforms the state of polarization without
substantially changing the degree of polarization.
By using both the calibration indicated by the steps 106 and 108
shown in FIG. 4 and calibration shown in FIG. 6, an absolutely
defined calibration frame of reference is established. That is, the
calibration corrects for all distortion of the state of
polarization downstream from the polarized optical source 9 or
9'.
The foregoing description is offered primarily for purposes of
illustration. While a variety of embodiments of a method and
apparatus for measuring polarization sensitivity of an optical
device under test and associated calibrations have been disclosed,
it will be readily apparent to those skilled in the art that
numerous other modifications and variations not mentioned above can
still be made without departing from the spirit and scope of the
invention as claimed below. For example, the focusing concave
mirror 12 shown in FIG. 2 can be replaced by a beam splitter and
associated collimating and focusing lenses. Furthermore,
measurement of reflection characteristics can be performed by
employing a beam splitter instead of the directional optical
coupler 100 shown in FIG. 5. Accordingly, the scope of the
invention can only be ascertained by reference to the appended
claims.
* * * * *